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The Heart of Special Relativity: Lorentz Transformation Equations

Special Relativity was first published in 1905 by Albert Einstein at age 26 working quietly in the Swiss Patent Office, Bern, Switzerland, under the title "On The Electrodynamics Of Moving Bodies", translated from "Zur Elektrodynamik bewegter Körper", Annalen der Physik, 17, 1905.

The Relativity Postulates: 

1). The Principle of Relativity - All the laws of physics in their simplest reduced form are transformable and hence invariant as between an infinite number of moving reference

systems ( inertial systems ), each one of which is moving uniformly and rectilinearly with respect to any other system and where no one system

is privileged or preferred over any other reference ( inertial ) system when measurements of length or time are taken.

2). The Principle of the Constancy of the Speed of Light - The speed of light in empty ( vacuo ) space is a universal constant as measured in any reference ( inertial ) system when

measured with rods and clocks of the same kind. This is always true notwithstanding any "relativistic effects" of either the

Lorentz length contraction or time dilation as earlier revealed by the Michelson-Morley Experiment (1887 ).

The Relativity Corollaries: 

a). The "luminiferous aether" does not exist. [ This entirely comports with the results of the Michelson-Morley Experiment. ]

b). The "Fallacy of Simultaneity at a Distance" - There is no such reality as simultaneously occurring events when measured at great distances or at velocities approaching

the speed of light, c . [ The proof of this will be given later in "Some Results of Lorentz Transformation Equations". ]

The Relativity Assumptions - The Cosmological Principle: 

i).  Isotropy -  space - time is uniform and symmetric in all ( x, y, z ) directions.

That is, there is no one preferred reference point or direction in space-time.

ii).  Homogeneity -  space - time possesses the quality of uniformity in structure and composition in all ( x, y, z ) directions.

That is, the geometry ( metric ) of space-time is the same from any point to any other point in the universe.

iii).  Systems -

S = S ( x ,y ,z , t ), inertial system at rest relative to S'

S' = S' ( x' ,y' ,z' , t' ), inertial system moving uniformly and rectilinearly along the x ( x' ) -axis with constant relative velocity v  with respect to inertial system S, where

x-axis is parallel and coincident or common to x'-axis

and likewise

y-axis, z-axis are respectively parallel to y'-axis, z'-axis. 

The Essential Relativity Problem: 

In order to understand an event E'  in relatively moving system S'  for an observer O  at the origin in stationary system S, we need to understand the rules of transformation for the following coordinates of this event E for an observer O'  at the respective origin in system S' :

That is, each P ( x, y, z, t ) space-time point in S  is invariably transformable to some other P' ( x', y', z', t' ) space-time point in S' .

The Solution to the above Essential Relativity Problem will give us The Lorentz Transformation Equations:

Therefore the differential form of the above equations becomes

or

for an array of unknown coefficients ( a₁₁ ... a₄₄ ) to whose solution essentially defines the task ahead to the Relativity Problem.

Because of space-time homogeneity all of the coefficients a_α,β  (α, β = 1,2,3,4) are independent of event E (E' ) coordinates ( x, y, z, t ), ( x', y', z', t' ) and therefore the equation set (2) is "integrable" and hence must be "linear transformation" equations.

Furthermore, because of space-time homogeneity all space-time points, P ( x, y, z, t ) in S and P' ( x', y', z', t') in S', are equivalent under linear transformation.

By calculus integration we get: 

So far so good. 

Let time t' = t = 0 at the instant the origin of ( 0,0,0 ) in S coincides with ( 0,0,0 ) in relatively moving S', then

Also because there is no relative motion in the y or z directions,

A little bit simpler, no?

By The Principle of Relativity and the invariant manner by which parallel lengths of rods in S and S' respectively are moving orthogonally ( at right angles ) to the relative direction of motion along the x ( x' ) -axis, it follows that they will not experience a Lorentz contraction along their y ( y' ) and z ( z' ) -axes.

Hence a rod of length 1 lying along the y-axis from y = 0 to y = 1 in S will also appear to possess a length of 1 in S' if this same rod is fixed along the y'-axis from y' = 0 to y' = 1. Likewise for a rod of length 1 along the z ( z' ) -axis.

This all implies 

And hence,

Also because of our basic space-time isotropy assumption (space-time is the same in all directions), t' and x' will not be dependent upon the y - and z - axes since any two ( or more ) S' -  clocks in the y'z' - plane placed symmetrically around the x - axis will appear to disagree as seen by an observer in S which would otherwise violate our isotropic assumption.

This all implies

which in turn implies the following reduced transformation equations:

Getting closer, getting closer.

Now event E'  in moving system S'  at origin O' , x' = 0, must also satisfy x = v t in system S where S'  is moving rectilinearly and uniformly away from system S with constant velocity v using the following constraints:

Again, our reduced transformation equations become:

Applying the 2nd Relativity Principle - The Principle of the Constancy of the Speed of Light - it must be that as S'  moves past S with a constant velocity v at time t' = 0 whose speeding origin O'  coincides exactly with origin O of system S at the precise moment, time t = t' = 0, for an event E , flash of light emanating at origin O' , x' =0, there will therefore be an expanding electromagnetic sphere of light propagating with constant speed c in all x, y, z directions in both S and S'  systems. Hence the speed of its progress in either system will be equal and can therefore be described by either transformation set of coordinates ( x, y, z, t ) or ( x', y', z', t' ) as follows: 

And the progress of the light propagation can be described by either equation:

⇒

,

since

Expanding and rearranging,

But equation ( 6b ) for moving system S'  must also satisfy the conditions of ( 6a ) for stationary system S. We therefore force this condition as follows:

Whew!

However we must continue ... 

We next solve these three simultaneous equations ( 7 ) by first eliminating as follows:

This entire "elimination process" can be viewed on Page "Solution to Equation ( 7a )" which is rather long and difficult. However the results are:

, 

where always the positive (+) sign of the square root is taken.

Have confidence that we are almost at the end! Faith, faith!!

We now substitute ( 7a ) equations into ( 5 ) as follows:

⇒

      .

These equations ( 8 )  are the famous Lorentz Transformation Equations which are integral to Special Relativity and thereby forms its mathematical basis.

At small values of v, v << c, where velocities are within the normal range of human experience ( excluding of course experiences of Quantum particle physicists, ha! ), Lorentz Transformation Equations easily reduce to traditional Galilean Transformation Equations as follows:

. 

Just to elucidate slightly more, Lorentz Transformation Equations as given above in ( 8 ) are those transformation equations where the observer O'  is standing in moving system S'  relative to stationary system S and attempting to derive his/her own coordinates ( x', y', z', t' ) relative to system S - i.e., as system S relatively "moves away".

The inverse of Lorentz Transformation Equations equations are therefore those transformation equations where the observer O is standing in stationary system S and is attempting to derive his/her coordinates ( x, y, z, t ) in S as system S'  relatively "moves away":

. 

And,

for small values of v, v << c.

See Page "Solution to Equation (9)" for this somewhat simpler derivation than that which is shown on Page "Solution to Equation (7a)".

We are thus finished!

These equations ( 8 ) - ( 9a ) are then the necessary tools for Relativity Mathematics and hence for Special Relativity cosmology. It is actually rather simple algebraic equations which form the basis of Special Relativity.

There are also other means and methods for deriving these Lorentz Transformation Equations such as partial differential geometry, etc., nevertheless the final result will always be the same as has already been derived. So why not stay with simple Algebra?

Continuing ...

Because Michelson and Morley [ see Michelson-Morley Experiment ( 1887 ) ]  were able to increase their fringe accuracy to within 1/100th of an interference fringe in their famous experiment, they were able to experimentally demonstrate the inadequacy of the traditional Galilean Transformation Equations to which even Isaac Newton took as ultimate truth.

Notice also that as S'  moves away from S, longitudinal velocity transforms from ( -v ) to ( +v ). This last observation is not trivial as neither element of ( x, t ) ∈ S is directly translatable to ( x', t' ) ∈ S' .

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